Frequency sources, often generated through oscillators, include noise that appears as a superposition of causally generated signals (deterministic) and random (nondeterministic) noise. The noise creates time-dependent phase and amplitude fluctuations in the frequency source. Measurements of these fluctuations characterize the frequency source in terms of amplitude modulation (AM) and phase modulation (PM) noise.
Frequency stability can be defined as the degree to which an oscillating source produces the same frequency throughout a specified period of time. A signal wave shape having a perfect sine function is considered to have the highest frequency stability. A given frequency stability will decrease as the signal wave holds anything different than the perfect sine function. For example, RF and microwave sources exhibit some amount of frequency instability.
The frequency stability can be divided into long-term stability and short-term stability. Long-term stability describes the frequency variations that occur over long time periods expressed in parts per million per hour, day, month or year. Short-term frequency stability contains all the elements causing frequency changes about the nominal frequency within less than a few seconds duration.
Phase noise is the term most widely used to describe the characteristic randomness of frequency stability. The spectral purity refers to the ratio of signal power to phase-noise sideband power.
Measurements of phase noise and AM noise may be performed in the frequency domain using a spectrum analyzer that provides a frequency window following a detector (double balanced mixer). Frequency stability can also be measured in the time domain with a gated counter that provides a time window following the detector.
There are two kinds of fluctuating phase terms for phase instability that may be used to characterize a real signal. They are deterministic and random. The deterministic terms are discrete signals appearing as distinct components in the spectral density plot. These discrete signals, often referred to as spurious, can be related to known phenomena in the signal source such as power line frequencies, vibration frequencies, and mixer products. The random kind of phase instability is commonly considered as phase noise. Thermal noise, shot noise, and noise of undetermined origin (such as flicker noise) are considered to be types of random noise. The source of random sideband noise in an oscillator includes these types of random noise.
Many terms exist to quantify the characteristic randomness of phase noise. The frequency or phase deviations of the source under test are usually measured in either the frequency or time domain. Since frequency and phase are related to each other, the terms that characterize phase noise are also related.
One fundamental description of phase instability (phase noise) is the spectral density of phase fluctuations on a per-Hertz basis. The spectral density defines the energy distribution as a continuous function, expressed in units of phase variance per unit bandwidth. Thus, the spectral density Sφ(fm) may be considered as equation (1)
                                          S            ϕ                    ⁡                      (                          f              m                        )                          =                                            Δ              ⁢                                                          ⁢                                                ϕ                  rms                  2                                ⁡                                  (                                      f                    m                                    )                                                                    BW              ⁢                                                          ⁢              used              ⁢                                                          ⁢              to              ⁢                                                          ⁢              measure              ⁢                                                          ⁢              Δ              ⁢                                                          ⁢                              ϕ                rms                                              ⁢                                    rad              2                        Hz                                              (        1        )            where BW (bandwidth) is negligible with respect to any changes in Sφ versus the Fourier frequency (or offset frequency) fm.
One useful measurement of noise energy is L(fm) which can be directly related to the spectral density Sφ(fm) by a simple approximation as shown in equation (2), which has generally negligible error if the modulation sidebands are such that the total phase deviations are much less than 1 radian.
                              L          ⁡                      (                          f              m                        )                          =                                            S              ϕ                        ⁡                          (                              f                m                            )                                2                                    (        2        )            
L(fm) is an indirect measurement of noise energy related to the RF power spectrum that can be observed on a spectrum analyzer. The U.S. National Bureau of Standards defines L(fm) as the ratio of the power in one phase modulation sideband to the total signal power (at an offset of fm Hertz away from the carrier). This relationship is shown by equation (3) below:
                                          L            ⁢                          (                              f                m                            )                                =                                                    P                ssb                                            P                s                                      ⁢                                                  ⁢                                                  =                                                                                                                              power                        ⁢                                                                                                  ⁢                        density                                                                                                                                                (                                                                                                                                            in                                ⁢                                                                                                                                  ⁢                                one                                ⁢                                                                                                                                  ⁢                                phase                                                                                                                                                                                                                                                                                                          ⁢                                                                  modulation                                  ⁢                                                                                                                                          ⁢                                  sideband                                                                                                                                                                    ⁢                                                                                                  )                                                                                                              total                  ⁢                                                                          ⁢                  signal                  ⁢                                                                          ⁢                  power                                            ⁢                                                          ⁢                                                          =                                                                                          single                      ⁢                                                                                          ⁢                      sideband                      ⁢                                              (                        SSB                        )                                            ⁢                                                                                          ⁢                      phase                                                                                                                                  noise                      ⁢                                                                                          ⁢                      to                      ⁢                                                                                          ⁢                      carrier                      ⁢                                                                                          ⁢                      ratio                      ⁢                                                                                          ⁢                      per                      ⁢                                                                                          ⁢                      Hz                                                                                                          ⁢                                                      (        3        )            Here, the phase modulation sideband is based on per Hertz of bandwidth spectral density and fm is the Fourier frequency or offset frequency.
The spectral density of frequency fluctuations (SΔf(fm)) is also used for quantifying short term frequency instability (phase noise). The spectral density defines the energy distribution as a continuous function, expressed in units of frequency variance per unit bandwidth and can be expressed as shown in equation (4).
                                          S                          Δ              ⁢                                                          ⁢              f                                ⁡                      (                          f              m                        )                          =                                            Δ              ⁢                                                          ⁢                                                f                  rms                  2                                ⁡                                  (                                      f                    m                                    )                                                                    BW              ⁢                                                          ⁢              used              ⁢                                                          ⁢              to              ⁢                                                          ⁢              measure              ⁢                                                          ⁢              Δ              ⁢                                                          ⁢                              f                rms                                              ⁢                                    Hz              2                        Hz                                              (        4        )            where BW is negligible with respect to any changes in Sφ versus fm.
A logarithmic plot of the spectral density of the phase modulation sideband is typically used for phase noise in the phase-frequency domain because of the large magnitude variations of the phase noise on an oscillator. These plots are typically expressed in dB relative to the carrier per Hz (dBc/Hz).
The spectral density Sφ(fm), the noise energy ratio L(fm), and the spectral density of the frequency fluctuations SΔf(fm) are related, and their relationships can be described using logarithmic terms as shown in equation (5) and equation (6) below.
                                                        S              ϕ                        ⁡                          (                              f                m                            )                                ⁡                      [                                          dB                ⁢                r                            Hz                        ]                          =                                                            S                                  Δ                  ⁢                                                                          ⁢                  f                                            ⁡                              (                                  f                  m                                )                                      ⁡                          [                              dBHz                Hz                            ]                                -                      20            ⁢                                                  ⁢            log            ⁢                                                            f                  m                                ⁡                                  (                  Hz                  )                                                            1                ⁢                                  (                  Hz                  )                                                                                        (        5        )                                                      L            ⁡                          (                              f                m                            )                                ⁡                      [                                          dB                ⁢                c                            Hz                        ]                          =                                                            S                                  Δ                  ⁢                                                                          ⁢                  f                                            ⁡                              (                                  f                  m                                )                                      ⁡                          [                              dBHz                Hz                            ]                                -                      20            ⁢                                                  ⁢            log            ⁢                                                            f                  m                                ⁡                                  (                  Hz                  )                                                            1                ⁢                                  (                  Hz                  )                                                              -                      3            ⁢                                                  ⁢            dB                                              (        6        )            where dBHz/Hz is dB relative to one Hz per Hz bandwidth, dBr/Hz is dB relative to one radian per Hz bandwidth, and dBc/Hz is dB relative to a carrier per Hz bandwidth.
The basic idea of an oscillator is to convert dc power to a periodic, sinusoidal RF output signal. Though all oscillators ultimately need a nonlinear description of their behavior, a linear approach is sufficient for their analysis and design. FIG. 1 illustrates a block diagram of an oscillator. The oscillator includes an amplifier with a frequency dependent gain G(jω) and a frequency dependent feedback network H(jω)). The feedback network includes a resonator circuit with a quality factor Q. The resonating circuit has losses due to the finite quality factor and can be modeled as a parallel RLC resonance circuit as shown in FIG. 2. This circuit is also called a tank circuit in the literature.
The inductor L and the capacitor C determine the resonance frequency and the resistor R represents the losses in the circuit. The resistor R determines the Q of the resonator. The impedance of the circuit looking at the input port can be described using equation (7).
                              Z          in                =                              (                                          1                R                            +                              1                                  j                  ⁢                                                                          ⁢                  ω                  ⁢                                                                          ⁢                  L                                            +                              j                ⁢                                                                  ⁢                ω                ⁢                                                                  ⁢                C                                      )                                -            1                                              (        7        )            
The resonance frequency can be established when the imaginary part of equation (7) is equal to zero. This means that the maximum amount of energy is oscillating between the inductor and capacitor. The oscillation frequency is shown below as equation (8).
                              1                                    ω              c                        ⁢            C                          =                              ω            c                    ⁢                                          ⁢          L                                    (        8        )            and the resonance frequency can be described as shown in equation (9).
                              ω          c                =                  1                      LC                                              (        9        )            
The Q is defined as the bandwidth BW of the resonance graph (see FIG. 3). The Q for a resonator with losses described by the resistance R can be modeled as shown in equation (10).
                    Q        =                              R                                          ω                c                            ⁢              L                                =                                    ω              c                        ⁢            CR                                              (        10        )            
The transfer function for a conventional oscillator block can be derived as equation (11):
                                          V            o                                V            in                          =                              G            ⁡                          (                              j                ⁢                                                                  ⁢                ω                            )                                            1            +                                          G                ⁡                                  (                                      j                    ⁢                                                                                  ⁢                    ω                                    )                                            ⁢                              H                ⁡                                  (                                      j                    ⁢                                                                                  ⁢                    ω                                    )                                                                                        (        11        )            
For an oscillator, V0 is nonzero when Vin is equal to zero, the oscillation condition can be extracted as shown in equation (12) and equation (13).
                                                                  G              ⁡                              (                                  j                  ⁢                                                                          ⁢                                      ω                    c                                                  )                                      ⁢                          H              ⁡                              (                                  j                  ⁢                                                                          ⁢                                      ω                    c                                                  )                                                              =        1                            (        12        )                                          arg          ⁡                      [                          G              ⁢                              (                                  j                  ⁢                                                                          ⁢                                      ω                    c                                                  )                            ⁢                              H                ⁡                                  (                                      j                    ⁢                                                                                  ⁢                                          ω                      c                                                        )                                                      ]                          =                  180          ⁢          °                                    (        13        )            
These magnitude and phase conditions have to be fulfilled at one frequency to get stable oscillation at the output of the oscillator.
Dating back to 1966, D. B. Leeson published a model for describing the output noise behavior of a feedback oscillator (see D. B. Leeson in “A simple model of feedback oscillator noise spectrum,” IEEE Proceedings, Vol. 54, February 1966). This model is still widely used for estimating the output spectral density of the phase noise of an oscillator. Referring again to FIG. 2, by assuming that the output is the voltage across the tank circuit, the only source of noise is the white thermal noise of the tank conductance. This noise can be represented as a current source across the parallel resonance circuit with a mean-square spectral density of equation (14).
                                                        i              n              2                        _                                Δ            ⁢                                                  ⁢            f                          =                              4            ⁢                                                  ⁢            kT                    R                                    (        14        )            
The current noise obtained through equation (14) becomes voltage noise when multiplied by the effective impedance looking into the current source. By considering that the energy restoration element must contribute an average effective negative resistance that precisely cancels the positive resistance of the parallel resonance circuit, the effective impedance looking into the noise current source is the same as the impedance of a perfectly lossless LC network. But at resonance, this is zero. For a relatively small offset frequency Δω from the center frequency ωc, the impedance of an LC resonance circuit is approximately described by equation (15).
                              Z          ⁡                      (                                          ω                c                            +                              Δ                ⁢                                                                  ⁢                ω                                      )                          ≈                              -            j                    ⁢                                                    ω                c                            ⁢              L                                      2              ⁢                                                          ⁢              Δ              ⁢                                                          ⁢                              ω                /                                  ω                  c                                                                                        (        15        )            
By using the definition of quality factor Q as described with respect to equation (10), the impedance of an LC resonance circuit yields equation (16).
                                                    Z            ⁡                          (                                                ω                  c                                +                                  Δ                  ⁢                                                                          ⁢                  ω                                            )                                                ≈                              -            R                    ⁢                                    ω              c                                                      2                ⁢                                                                  ⁢                Q                ⁢                                                                  ⁢                Δ                ⁢                                                                  ⁢                ω                            ⁢                                                                                                      (        16        )            
Then, the spectral density of the mean-square noise voltage can be obtained by multiplying the spectral density of the mean-square noise current with the squared magnitude of the impedance of an LC resonance circuit as shown in equation (17).
                                                        v              n              2                        _                                Δ            ⁢                                                  ⁢            f                          =                                                                              i                  n                  2                                _                                            Δ                ⁢                                                                  ⁢                f                                      ⁢                                                          Z                                            2                                =                      4            ⁢                                                  ⁢                                          kTR                ⁡                                  (                                                            ω                      c                                                              2                      ⁢                                                                                          ⁢                      Q                      ⁢                                                                                          ⁢                      Δ                      ⁢                                                                                          ⁢                      ω                                                        )                                            2                                                          (        17        )            
The power spectral density of the output noise is frequency-dependent. This 1/f2 behavior represents two characteristics. The first is that the voltage frequency response of an LC resonance circuit rolls off as 1/f to either side of the center frequency. The second is that the power is proportional to the square of voltage. An increase of an LC resonance circuit's Q reduces the noise density with all other parameters constant.
Thermal noise causes fluctuations in both amplitude and phase as shown through equation (17) above. Noise energy would split equally into amplitude and phase noise if not for the amplitude limiting that occurs in real circuits. The amplitude limiting mechanisms present in all practical oscillators result in attenuated amplitude noise. In order to quantify this noise level, it is conventional to normalize the mean-square noise voltage density to the mean-square carrier voltage in decibels. This normalization expresses the following phase noise equation (18).
                                                                                          L                  ⁡                                      (                                          f                      m                                        )                                                  ⁡                                  [                                                            dB                      ⁢                      c                                        Hz                                    ]                                            =                              10                ⁢                                                                  ⁢                log                ⁢                                                      P                    noise                                                        P                    carrier                                                  ⁢                                                                  ⁢                per                ⁢                                                                  ⁢                Hz                                                                                        =                              10                ⁢                                                                  ⁢                                  log                  ⁡                                      [                                                                                            2                          ⁢                                                                                                          ⁢                          kT                                                                          P                          carrier                                                                    ⁢                                                                        (                                                                                    ω                              c                                                                                      2                              ⁢                                                                                                                          ⁢                              Q                              ⁢                                                                                                                          ⁢                              Δ                              ⁢                                                                                                                          ⁢                              ω                                                                                )                                                2                                                              ]                                                                                                          (        18        )            
However, equation (18) requires many simplifying assumptions. Therefore, there are some significant differences between the spectrum obtained by equation (18) and the real oscillator spectrum. To solve this discrepancy, Leeson provided a modification to equation (18) as shown in equation (19).
                                          L            ⁡                          (                              f                m                            )                                ⁡                      [                                          dB                ⁢                c                            Hz                        ]                          =                  10          ⁢                                          ⁢                      log            ⁡                          [                                                                    2                    ⁢                    F                    ⁢                                                                                  ⁢                    kT                                                        P                    carrier                                                  ⁢                                                      (                                          1                      +                                                                        ω                          c                                                                          2                          ⁢                                                                                                          ⁢                          Q                          ⁢                                                                                                          ⁢                          Δ                          ⁢                                                                                                          ⁢                          ω                                                                                      )                                    2                                ⁢                                  (                                      1                    +                                                                  Δ                        ⁢                                                                                                  ⁢                                                  ω                                                      1                            /                                                          f                              3                                                                                                                                                                                                    Δ                          ⁢                                                                                                          ⁢                          ω                                                                                                                              )                                            ]                                                          (        19        )            
These modifications consist of a factor F to account for the increased noise in the 1/(fm)2 region, an additive factor of unity to account for the noise floor, and a multiplicative factor to provide a 1/|fm|3 behavior at sufficiently small offset frequency. Although Leeson's model is useful for obtaining intuitive insight, the factor F is an empirical Fitting parameter and must be determined from measurement, diminishing the predictive power of the phase noise equation. Also, the 1/f corner of device noise is not precisely equal to (fm)1/f3 in practice.
There are many methods for measuring phase noise. Three common methods are direct measurement, PLL-based measurement (two oscillator method), and FM discriminator (delay-line based) measurement (one oscillator method).
The most simple and straightforward method of phase noise measurement is direct measurement. That is, to input the test signal into a spectrum analyzer arid directly measure the power spectral density of the oscillator. However, this method may be significantly limited by the spectrum analyzer's dynamic range, resolution, and LO (local oscillator) phase noise. Though this direct measurement is not useful for measurements close in to a drifting carrier, it is convenient for qualitative quick evaluation on sources with relatively high noise. The measurement is valid if the following conditions are met. The first is that the spectrum analyzer SSB phase noise at the offset of interest must be lower than the noise of the Device-Under-Test (DUT). The second condition is that since the spectrum analyzer will measure total noise power, the amplitude noise of the DUT must be significantly below its phase noise (Typically 10 dB will suffice).
FIG. 4 shows a typical display of an oscillator mixed down to DC. The main advantage of this method is its simple test set-up and that it can measure phase noise at high offset frequencies from the carrier. However, there are several disadvantages. One is that the spectrum analyzer cannot distinguish a difference between amplitude noise and phase noise and one does not have any idea regarding the noise power in the amplitude and phase of a DUT. Finally, some correction factors have to be incorporated in order to compensate the phase noise power since the phase noise power is normalized to a bandwidth of 1 Hz in an ideal rectangular filter but the resolution bandwidth filter of the spectrum analyzer is non-ideal.
The Phase Locked Loop (PLL)-based method is one of the most sensitive methods for measuring phase noise. Two oscillators send signals to the two RF ports of a mixer. The IF signal of a mixer passes through a low pass filter to keep out the sum frequency components and then sends them back in a small bandwidth's signal to lock one oscillator to the other. The fundamental block diagram is shown in FIG. 5. The basis of this method is the double-balanced mixer used as a phase detector.
Two signals at identical frequencies and nominally in phase quadrature (i.e. 90° out of phase) are input to the phase detector (a double balanced mixer). At quadrature, the output of the phase detector is a difference frequency of 0 Hz and an average voltage output of 0 V. There is a small fluctuation voltage, ΔV. For small phase deviations (Δφ<<1 rad), this fluctuating voltage is proportional to the fluctuating phase difference between the two signals. This phase difference represents the combined phase modulation sidebands of the two input signals. When the two input signals are identical in frequency and in phase quadrature, the output of the phase detector is a voltage directly proportional to the combined phase modulation sidebands of the two input signals.
The frequency and amplitude offsets are then removed such that the two input signals are again at identical frequencies, and are set in phase quadrature. It is important to use the mixer in its linear region where the voltage output is directly proportional to the phase difference of the input signals by a constant A (i.e. the mixer's efficiency).
FIG. 6 shows a typical mixer-phase detector characteristic. The mixer produces an output voltage V(t) proportional to the fluctuating phase difference between the two input signals φLO−φRF. The point of maximum phase sensitivity and the center of the region of most linear operation occur where the phase difference between the two inputs is equals to 90° or phase quadrature.
In this measurement method, the phase quadrature is the point of maximum phase sensitivity and the region of most linear operation. Any small deviation from quadrature results in a measurement error. Table 1 shows the typical error table of PLL based phase noise measurement system (Agilent 11729B phase noise measurement system).
TABLE 1Error contribution to the measurementOffset from quadratureMeasurement Error1°−0.001 dB 3°−0.01 dB10° −0.13 dB
The PLL-based method has several advantages and disadvantages. The PLL-based method uses a smaller spectrum analyzer dynamic range after converting the RF signal to the baseband signal. The internal noise of the spectrum analyzer is not the limiting factor. A low noise preamplifier is used to amplify the baseband signal to meet the range of the spectrum analyzer. In addition, the mixer operating as a phase detector is suppressing the amplitude noise due to its quadrature input condition in this setup. Currently, good mixers achieve an AM noise suppression from 30˜40 dB. For the PLL-based method, the measurement result is 3 dB higher for the case of two identical sources because the DUT and the reference have the same characteristic. The reference source is a high stable oscillator, which is the limiting factor of the test setup. A disadvantage for the PLL-based method is the need for two sources in this test setup.
Delay line discriminators are only capable of measuring phase based random noise, and are in fact insensitive to AM noise. This can be an important advantage when measuring the phase noise of sources which do have significant AM noise. Accordingly, it can be important to first identify the two types of noise present in a frequency source: AM noise and phase noise. Phase noise, generally considered to be the dominant form of random noise, is defined as the noise generated from random fluctuations in the phase of a frequency source. AM noise is simply the noise generated from random fluctuations in the amplitude of a frequency source.
Unlike the PLL based method, a frequency discriminator method (delay line method) does not require a second reference signal phased locked to a DUT. This makes the frequency discriminator method extremely useful for measuring DUTs that are difficult to phase kick. It can also be used to characterize sources with high-level, low-rate phase noise, or high close-in spurious sidebands, which can impose serious problems for the PLL-based method.
The delay line implementation of the frequency discriminator converts short-term frequency fluctuations (Δf) of DUT into voltage fluctuations (ΔV) that can be measured using a baseband analyzer. The conversion is two part process, first converting the frequency fluctuations into phase fluctuations and then converting the phase fluctuations to voltage fluctuations as shown in FIG. 7.
The frequency fluctuation to phase fluctuation transformation (Δf→Δφ) takes place in the delay line. The nominal frequency arrives at the double-balanced mixer at a particular phase. As the frequency changes slightly, the phase shift incurred in the fixed delay time will change proportionally. The delay line converts the frequency change at the line input to a phase change at the line output when compared to the non-delayed signal arriving at the mixer in the second path. The double-balanced mixer transforms the instantaneous phase fluctuations into voltage fluctuations (Δφ→ΔV). With the two input signals 90° out of phase, the voltage output is proportional to the input phase fluctuations. The power spectral density of the phase noise measured by this system can be described by equation (20).
                              L          ⁡                      (                          f              m                        )                          =                                                            V                                  out                  ,                  rms                                2                            ⁡                              (                                  f                  m                                )                                                    2              ⁢                                                (                                                            K                      ϕ                                        ⁢                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                                          f                      m                                        ⁢                                          τ                      d                                                        )                                2                                              ⁢                      (                          in              ⁢                                                          ⁢              1              ⁢                                                          ⁢              Hz              ⁢                                                          ⁢              measurement              ⁢                                                          ⁢              bandwidth                        )                                              (        20        )            where Kφ is a constant related to the output voltage of the signals passing through the delay line and the phase shifter.
As shown in equation (20), the system sensitivity is closely related to delay time τd. As the delay time increases, the sensitivity is better. In order to get a proper sensitivity, the delay line should be long enough since the delay time is inversely proportional to the phase noise.
Accordingly, means and methods for phase noise measurement continue to require additional research and improvement. In addition, there exists a need in the art for a measurement system that can be incorporated on-chip.